Detecting very low frequency magnetic fields based on precession modulation of magnetic field sensors

ABSTRACT

A very low frequency (VLF) magnetic field sensor, using spins sourced from insulating ferrimagnetic materials or ferrites, which can achieve high sensitivities competitive with modern sensors while simultaneously maintaining a small size, low power consumption, simplicity of design, and low cost. The magnetic field sensor utilizes nonlinear resonant precession modulation (RPM) dynamics of subatomic spins to attain parametric amplification of a magnetic field.

This application claims priority to, and the benefit of, U.S.provisional patent application Ser. No. 63/027,816 filed on May 20,2020, incorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

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BACKGROUND 1. Technical Field

The technology of this disclosure pertains generally to magnetic fieldsensors, and more particularly to employing nonlinear precessiondynamics of subatomic spins to attain parametric amplification of amagnetic field.

2. Background Discussion

Magnetic field sensors have been one of the cornerstone technologies inadvancing the progress of research and development in many areas. Fromgeophysics to biomedicine to communications, the collective domain overwhich magnetic field sensing and detection spans is incredibly broad andthe specific applications for its use are extremely diverse.

However, these current magnetic sensor technologies representsignificant tradeoffs between sensitivity and practicality.

Accordingly, the present disclosure provides a magnetic sensortechnology which can provide high levels of sensitivity while remainingpractical in terms of size, fabrication and cost factors.

BRIEF SUMMARY

This disclosure describes a sensor for detecting weak very low frequency(VLF) magnetic fields, as well a very low frequency (VLF) antenna, basedon spin precession modulation (SPM). The sensor and antenna exhibit highsensitivity and are practical for use in many applications in whichdetection of low level signals is required.

Further aspects of the technology described herein will be brought outin the following portions of the specification, wherein the detaileddescription is for the purpose of fully disclosing preferred embodimentsof the technology without placing limitations thereon.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The technology described herein will be more fully understood byreference to the following drawings which are for illustrative purposesonly:

FIG. 1A through FIG. 1D are diagrams which illustrate ferritemagnetization precession in the presence of a magnetic bias field withboth static and time varying components, as utilized according to atleast one embodiment of the present disclosure.

FIG. 2A through FIG. 2C are schematics of magnetic field sensoroperation using resonant precession modulation (RPM) principlesaccording to at least one embodiment of the present disclosure.

FIG. 3 are plots comparing theory versus simulations of magnetic fieldsensor operation using resonant precession modulation (RPM) principleswhich show high correlation between theory and the simulations.

FIG. 4 is a schematic diagram of an RPM based magnetic field sensorimplementation according to at least one embodiment of the presentdisclosure.

FIG. 5 are spectrum plots of power coupled to the inductive detectionloop for a simulation of an RPM sensor according to at least oneembodiment of the present disclosure, showing spectrum for driving coilvoltages of 45 mV and 91 mV.

FIG. 6 is a schematic of a setup for measurement of sensor sensitivityof an RPM sensor according to an embodiment of the present disclosure.

FIG. 7 is an equivalent circuit model of an experimental RPM sensoraccording to at least one embodiment of the present disclosure.

DETAILED DESCRIPTION 1. Introduction

High sensitivity magnetic field sensing is becoming increasinglyimportant in today's technological landscape from geophysics tobiomedicine to communications. In view of this a broad range of magneticsensor types have been proposed or developed.

There consequently has arisen an almost equally diverse range ofapproaches to magnetic field sensor design in terms of foundationalprinciples of operation. For example, super quantum interference device(SQUID) sensors operate based on the quantization of magnetic flux,optically pumped sensors operate based on atomic magneto-optic effects,induction sensors operate based on Faraday's law of induction, andmagnetic tunnel junction sensors operate based on polarization dependentelectron tunneling.

One defining characteristic of a magnetic field sensor issensitivity—the lowest field value that can be detected above the noisefloor in a unit of output bandwidth. A higher sensitivity, or theability to detect weaker fields, is crucial in modern applications whereworking with weak fields is the overwhelming norm. The sensitivity isstrongly dependent on the principle of operation of a sensor; however,the fact that there are still so many sensor types used today clearlyindicates that there are other important characteristics that must alsobe considered. Specifically, it tends to be the case that principlesyielding higher sensitivity have tradeoffs with regards to sensor sizeor power consumption that limit practicality and cost of implementation.

The tradeoff between sensitivity and practicality may be betterillustrated by examining various principles of operation. SQUID magneticfield sensors are widely recognized to be the most sensitive devices,able to achieve sensitivities better than 1 fT/Hz^(1/2); however, theyrequire cryogenic cooling which leads to a system that is large andcostly in terms of construction and power consumption. Optically pumpedsensors can be competitive with SQUIDs in sensitivity, but are afflictedby the requirement of optically pumped, heated atomic vapor cells whichcontribute again to a large and costly system that is complex to designand manufacture.

Optically pumped sensors are capable of achieving sub-pT/Hz^(1/2)sensitivities in the kHz to MHz frequency range but require a sensorvolume of approximately 57 cubic millimeters and power consumption onthe order of Watts. Inductive sensors boast zero power consumption whilealso being able to attain sensitivities better than 1 fT/Hz^(1/2);however, reaching these sensitivities generally requires the device tobe prohibitively large or heavy. The use of inductive sensors canachieve sub-fT/Hz^(1/2) sensitivity in the hundreds of Hz to hundreds ofkHz frequency range but require a sensor area of more than 24 squaremeters.

Magnetic tunnel junctions can be made extremely small; however, theyconsume moderate amounts of power and achieve moderate sensitivities.The use of an MgO based magnetic tunnel junction sensor was constructedwith a sensor area of just 626 μm², but there is only 97 pT/Hz^(1/2)sensitivity below 500 kHz as well as a power consumption of around −7dBm (˜0.1995 mW).

In contrast, the present disclosure introduces a novel approach thatallows for the realization of sensitive magnetic field sensors operatingin the low frequency (LF) and very low frequency (VLF) bands that alsomaintain a high degree of practicality. A preliminary design exhibits asensitivity threshold of less than 27 pT/Hz^(1/2) flat in these bands.While this design has only a moderately high sensitivity, it boasts asensor volume of only 0.053 cubic millimeters (mm³) and a very low powerconsumption of just −40 dBm (˜1e⁻⁷ Watts).

The approach to magnetic field sensing described herein was inspired byoptical magnetometry which exploits the nonlinearities of electron spinprecession dynamics to achieve parametric amplification and willhereafter be referred to as resonant precession modulation (RPM).

Operation in the VLF and LF bands encompass applications that includeunderground and underwater communications, space plasma research, orlow-cost magnetic resonance imaging.

The following sections will first cover the basic principles of RPM andderive analytical expressions describing RPM based sensors. Theadvantages of these sensors are then discussed along with an approachfor practical implementation. Finally, the setup and results arepresented for both simulations and a preliminary experiment.

2. Principles of Operation

FIG. 1A through FIG. 1D illustrate aspects 10, 30, 40 and 50 ofmagnetization dynamics of a ferrite material.

In FIG. 1A is illustrated 10 a ferrite material 12 exhibiting ferritemagnetization M(t) precession 18, with ferrite magnetization M_(t)(t)20, seen in the presence of a magnetic bias field with both a static H₀14 and time varying h_(s)(t) 16 component.

In FIG. 1B is shown a time domain plot 30 qualitatively depictingh_(s)(t) as well as a component of the ferrite magnetization M_(t)(t)transverse to the bias field for the case where h_(s)(t) is constant.

In FIG. 1C is shown 40 a similar plot depicting both h_(s)(t) andM_(t)(t) for the case where h_(s)(t) is varying sinusoidally.

In FIG. 1D is illustrated 50 a thin film ferrite material 52 exhibitingferrite magnetization precession M(t) 58, with M_(t)(t) 60, in thepresence of a magnetic bias field with both a static H₀ 54 and timevarying h_(s)(t) 56 component. Demagnetization effects cause theprecession to resemble a pendulum oscillation.

Although inspired by optical magnetometry, an extension of spindynamics-based magnetic field sensors requires significant changes torectify those aspects which contribute to the unsuitability of opticalmagnetometers for use in many magnetic field sensing applications.Perhaps foremost among these changes involves the medium from which thespins are sourced. Optical magnetometers use spins sourced from atomicvapors, the glass containers of which are the bottleneck preventingreductions in cost and device size. In contrast, the present disclosurefocuses instead on spins sourced from insulating ferrimagneticmaterials, or ferrites. The insulating nature, low magnetic loss at highfrequencies, and significant saturation magnetization values of ferritesare all imperative toward achieving high performance magnetic fieldsensors. Some of these attributes motivated inductive coil antennasusing ferrites, although of course these never employed the use ofnonlinear spin precession dynamics or other aspects of the operatingprinciples of the present disclosure. Ferrites additionally have anextensive history of incorporation in general high frequency electronicsresulting in now well-established processing methods that allow thematerial to be produced relatively inexpensively, which can be leveragedwhen implementing the teachings of the present disclosure.

Using ferrites as the selected medium, analysis of spin dynamics may beaccomplished through application of micromagnetic theory. This theory isable to practically describe the average behavior of the large ensembleof spins of a macroscopic ferrite material and is formulated classicallyin terms of the spatially and temporally dependent materialmagnetization M(r,t) based on a continuum approximation. References tomagnetization as opposed to spin will consequently be made for allfollowing analyses for the purposes of consistency, where it isrecognized that either may be immediately determined based on knowledgeof the other. From micromagnetic theory, the magnetization dynamics of aferrite material are governed by the Landau-Lifshitz-Gilbert (LLG)equation:

$\begin{matrix}{\frac{d{M\left( {r,t} \right)}}{dt} = {{{- \gamma}{\mu_{0}\left( {{M\left( {r,t} \right)} \times {H_{e}\left( {r,t} \right)}} \right)}} + {\frac{\alpha}{M_{S}}\left( {{M\left( {r,t} \right)} \times \frac{d{M\left( {r,t} \right)}}{dt}} \right)}}} & (1)\end{matrix}$where γ is the gyromagnetic ratio, μ₀ is the permeability of free space,H_(e)(r,t) is an effective magnetic field representing actual magneticfields as well as various other physics, α is the Gilbert dampingcoefficient representing magnetic loss, and M_(S)=|M(r,t)| is thesaturation magnetization. For purposes of simplicity, the assumptionwill be made initially that H_(e)(r,t) is spatially uniform with thedominant contributor to its value being some constant magnetic field H₀.It will further be assumed that the magnitude |H_(e)(r,t)|=H_(e)(r,t) issufficiently large such that M(r,t) is also spatially uniform, and thatmagnetic loss is neglected.

From the LLG equation (1), the means by which nonlinear spin precessiondynamics may be used to characterize magnetic fields of electromagneticwaves is elucidated. The equation indicates the equilibrium state ofM(t) to be aligned with H_(e), where it is noted that spatial andtemporal dependencies of these and all following variables will beomitted as appropriate. Any perturbation from the equilibrium state willresult in a continuous precession of M(t) about H_(e), as visualized inFIG. 1A and FIG. 1B, at the resonance frequency f₀ given by:

$\begin{matrix}{{f_{0} = {\frac{1}{2\pi}\gamma\mu_{0}H_{e}}}.} & (2)\end{matrix}$

If magnetic loss is accounted for, the precession will dampen and M(t)will eventually spiral back to its equilibrium state. Equation (2)contains the essence of optical magnetometry operations in which it maybe seen that, by determining f₀ through some means, an unknown H_(e) canbe directly computed. While such an approach is not readily extendedsuitably for receiving antenna operation, equation (2) also allows thecorrect presumption to be made that a time varying effective fieldmagnitude, say due to an additional magnetic field h_(s)(t) contributingto the effective field such that H_(e) (t)=H₀+h_(s)(t), will result in atime varying precession frequency.

This nonlinear behavior, termed a resonant precession modulation (RPM),is visualized in FIG. 1C and is the key to achieving the desiredmagnetic field characterization. Framed in more practical terms,consider the precession of M(t) for a ferrite biased with a sufficientlylarge magnetic field H₀. A dynamic, spatially uniform magnetic fieldh_(s)(t) is imposed on the ferrite and is polarized such that h_(s)(t)is parallel to H₀, and will result in a time varying H_(e) (t). Thismagnetic field thus manifests as a frequency modulation of theprecession, a detection of which then allows for ensuing extraction ofall the information that can be known about both the magnitude and phaseof the dynamic field.

3. Embodiments

FIG. 2A through FIG. 2C illustrate example embodiments 70, 90 and 110 ofmagnetic field sensor operation.

In FIG. 2A is illustrated an inductive detector 74 surrounding ferritematerial 72 shown performing magnetic field sensing under the principlesof RPM depicting ferrite magnetization M(t) 80 precession, with M_(t)(t)82 where orientations of the bias magnetic field H₀ 76, the perturbingmagnetic field H_(p)(t) 78, the incident magnetic field H_(s)(t) 84.

In FIG. 2B is illustrated 90 an inductive coil antenna 92 operationunder conventional principles showing H_(s)(t) 94.

In FIG. 2C is illustrated a ferrite rod (core) 112 loaded inductive coil114 operation where positioning of the ferrite core within the inductivedetector is visualized, with ferrite magnetization M(t) and magneticfield H_(s)(t) 118 is shown.

3.1. Development of Analytical Expressions

A theoretical analysis of RPM details explicitly the manner in whichmagnetic field information is infused in spin dynamics, which providesthe foundation for which advantages of magnetic field sensor operationbased on RPM may be illuminated, and generates expressions that areverifiable through simulation.

Considering the scenario visualized in FIG. 2A in which a ferritematerial is biased with a static magnetic field H₀{circumflex over (z)}and a dynamic magnetic field H_(s)(t){circumflex over (z)} is incidentat the material. As previously, the assumptions are made that both H₀and the wavelength of the incident wave are sufficiently large. Unlikethese previous descriptions, however, magnetic loss will not beneglected and so a perturbing magnetic field H_(p)(t){circumflex over(z)} at the frequency f₀ defined in equation (2) exists to maintainmagnetization precession with a constant angle θ from its equilibriumstate amid the effects of damping. This perturbing field is assumed tobe sufficiently small such as to not contribute significantly toH_(e)(t). Then M_(t)(t) is a component of M(t) transverse to the biasfield, and may be written from equation (1) as:

$\begin{matrix}{{M_{t}(t)} = {M_{s}{\sin(\theta)}{\cos\left( {{2\pi f_{0}t} + {\int\limits_{t - \tau}^{t}{\gamma\mu_{0}{h_{s}(t)}dt}}} \right)}}} & (3)\end{matrix}$where τ=(2πf₀α)¹ which is the characteristic decay time of resonantprecession. Describing the magnetic field of the incident wave asH_(s)(t)=H_(s) cos(2πf_(s)t+ϕ), equation (3) can be evaluated to yield:

$\begin{matrix}{{M_{t}(t)} = {M_{s}{{\sin(\theta)}\left\lbrack {{\cos\left( {2\pi f_{0}t} \right)} - {\frac{1}{2}\gamma\mu_{0}H_{s}\tau{\sin\left( {{2{\pi\left( {f_{0} + f_{s}} \right)}t} + \phi} \right)}} - {\frac{1}{2}\gamma\mu_{0}H_{s}\tau{\sin\left( {{2{\pi\left( {f_{0} - f_{s}} \right)}t} - \phi} \right)}}} \right\rbrack}}} & (4)\end{matrix}$under the conditions of γμ₀H_(s)τ<<1 and 2πf_(s)τ<<1. It may be seenfrom this equation that the effect of the wave is to generate terms inthe precession dynamics at the sum and difference frequencies f₀±f_(s),and that a detection of M_(t)(t) theoretically allows all theinformation about the dynamic magnetic field to be extracted from theseterms.

Equation (4) can be applied to analyze the amplification of the magneticfield information achieved by RPM as compared to those operating underconventional principles. In order to realize the necessary transductionto an electrical signal, an inductive detection of M_(t)(t) is assumedusing a simple conductive coil of N turns and area A oriented as shownin FIG. 2A. The amplitude V_(RPM) of the open circuit voltage inducedacross the terminals of the coil, are effectively representative of thestrength of the transduced signal, by a term of M_(t)(t) at f₀±f_(s) canbe found using the principles of Faraday induction to be:V _(RPM)=πNAμ₀ ² M _(s) sin(θ)γH _(s)τ(f ₀ ±f _(s)).  (5)

The same coil operating under conventional antenna principles asvisualized in FIG. 2B, where the open circuit voltage amplitude V_(con)across its terminals, is now induced directly by the incident magneticfield and so in order to achieve the largest possible voltage amplitude,the coil is oriented with its axis parallel to the magnetic field.Again, applying the principles of Faraday induction, this amplitude isgiven by:V _(con)=2πNAμ₀ H _(s) f _(s).  (6)

Taking the ratio of equation (5) to equation (6), the signalamplification achieved by a receiving antenna based on RPM. as opposedto one operating under conventional principles, is found to be:

$\begin{matrix}{\frac{V_{RPM}}{V_{con}} = {\frac{M_{S}{\sin(\theta)}}{\Delta H}\left( \frac{f_{0}}{f_{s}} \right)}} & (7)\end{matrix}$where

${\Delta\; H} = \frac{2}{\mu_{0}{\gamma\tau}}$is the linewidth of resonant precession.

Equation (4) can be further applied to derive a metric independent ofprecession angle θ that will later be used to evaluate the validity ofthe developed theory through a comparison with simulation results.Specifically, this metric is the amplitude ratio of the M_(t)(t) term ateither the sum or difference frequency f₀±f_(s) to that at theunmodulated precession frequency f₀, expressed analytically as:

$\begin{matrix}{{\frac{M_{t}^{f_{0} \pm f_{s}}}{M_{t}^{f_{0}}}} = {\frac{H_{s}}{H_{0}}Q}} & (8)\end{matrix}$where Q=πf₀τ is the quality factor of the resonant precession. ApplyingFaraday's law, this amplitude ratio can also be found from the opencircuit RMS voltages as

$\begin{matrix}{\frac{V_{RPM}}{V_{0}} = {{\frac{M_{t}^{RPM}}{M_{t}^{0}}} = {\frac{H_{s}}{H_{0}}Q}}} & (9)\end{matrix}$where V₀ corresponds to the magnetization term at the resonancefrequency.

The sensitivity of RPM sensors can be quantified recognizing that noiseis dominated by ferrite damping noise. This noise can be modeled with athermal resistor R and does not depend on the pumping power injected tomaintain precession. A full circuit model for an RPM sensor is providedin FIG. 7 in the Supplementary section. Sensitivity then corresponds tothe RMS field of interest H_(s)/√{square root over (2)}, where H_(s) issuch thatV _(RPM)=√{square root over (4kTR)}  (10)

Here, k is the Boltzmann constant and T is temperature. Applyingequation (10) to equation (11), an explicit expression for sensitivitycan be derived as,

$\begin{matrix}{{\frac{H_{s}}{H_{0}}QV_{0}} = \sqrt{4kTR}} & (11)\end{matrix}$

By convention, sensitivity is typically expressed in terms of themagnetic flux densities and so it will be denoted as δB_(s), given as

$\begin{matrix}{{\delta\; B_{s,{RPM}}} = {\frac{\mu_{0}H_{0}}{\sqrt{2}} = {{\frac{\mu_{0}H_{0}}{Q}\sqrt{\frac{2kTR}{V_{0}^{2}}}} = {\mu_{0}\Delta H\sqrt{\frac{2kT}{P_{p}}}}}}} & (12)\end{matrix}$where P_(p)=V²/R is the pumping power consumed by the ferrite. It shouldbe noted that the two sideband frequencies of equation (5) can becoherently combined with demodulation circuitry, resulting in anadditional factor of 1/√{square root over (2)} in the sensitivity ofequation (12).

Previously, the effective field H_(e) was described as encompassing theeffects of magnetic fields as well as various other physics on themagnetization. These other physics are represented with additivecomponents to H_(e) and many can be neglected for a biased ferrite.

One component that may be significant however, is the demagnetizationfield. The demagnetization field accounts for the dipole-dipoleinteraction between spins and is dependent on the ferrite shape, size,and biasing. Of particular interest in this work are ferrites with athin-film geometry, which lend themselves well to miniaturization andcan be produced with high quality through mature processing methods. Foran RPM sensor employing a thin film ferrite biased in-plane, thedemagnetization field is indeed significant, necessitating modificationsto the above equations. Magnetization precession in this caseapproximately resembles a pendulum oscillation, as visualized in FIG.1D, with a resonance frequencyω₀=μ₀γ√{square root over (H ₀(H ₀ +M _(s)))}.  (13)

With the addition of the same time-harmonic h_(s)(t), |h_(s)(t)|<<Hpolarized along the bias field, a Taylor expansion can be used toexpress the time varying precession frequency as

$\begin{matrix}{{\omega(t)} = {\omega_{0} + {\frac{\mu_{0}{\gamma\left( {{2H_{0}} + M_{s}} \right)}}{2\sqrt{H_{0}\left( {H_{0} + M_{s}} \right)}}{{h_{s}(t)}.}}}} & (14)\end{matrix}$

The precession time constant τ and the quality factor Q are alsomodified by the demagnetization field, with τ=2Q/ω₀ and

$\begin{matrix}{{Q = \frac{\sqrt{H_{0}\left( {H_{0} + M_{s}} \right)}}{\alpha\left( {{2H_{0}} + M_{s}} \right)}}.} & (15)\end{matrix}$

Equation (15) can then be re-written as,

$\begin{matrix}{{\omega(t)} = {\omega_{0} + {\frac{\mu_{0}\gamma}{2\alpha Q}{h_{s}(t)}}}} & (16)\end{matrix}$

Starting with equation (17) below, instead of equation (3), the exactsame procedure taken to derive equations (5) through (13) can be appliedto derive the analogous equations corresponding to an RPM sensor with athin film ferrite. The open circuit RMS voltage for detection of one ofthe sidebands is

$\begin{matrix}\begin{matrix}{V_{RPM} = {\frac{1}{2\alpha\; Q}\frac{1}{2\sqrt{2}}\mu_{0}^{2}\gamma H_{s}\tau\; N\; A\;\omega_{0}M_{s}{\sin(\theta)}}} \\{= {\frac{1}{2\sqrt{2\;\alpha}}\mu_{0}^{2}\gamma H_{s}NAM_{s}{\sin(\theta)}}}\end{matrix} & (17)\end{matrix}$

The amplification, amplitude ratio, and single sideband sensitivity arefound to be

$\begin{matrix}{\frac{V_{RPM}}{V_{ind}} = {\frac{M_{S}{\sin(\theta)}}{\Delta H}\left( \frac{\omega}{\omega_{s}} \right)}} & (18) \\{\left| \frac{M_{t}^{RPM}}{M_{t}^{0}} \right| = {{\frac{\left( {{2H_{0}} + M_{s}} \right)}{2{H_{0}\left( {H_{0} + M_{s}} \right)}}H_{S}Q} = \frac{H_{s}}{\Delta H}}} & (19) \\{{\delta\; B_{s,{RPM}}} = {\mu_{0}\Delta H\sqrt{\frac{2kT}{P_{p}}}}} & (20)\end{matrix}$where the linewidth is still defined as ΔH=2αω₀/μ₀γ. Comparing equation(18) through equation (20) to equations (6), (8) and (9), it can beconcluded that, for a given ferrite material, RPM sensor performance isindependent of whether or not the demagnetization field is significant,even though the resonance frequency and quality factor are modified bythe demagnetization.

FIG. 3 illustrates an example embodiment 130 comparing between theoryand simulation of RPM operation, showing a plot of the transversemagnetization amplitude ratio for a component at the sum or differencefrequency f₀±f_(s) to a component at the unmodulated precessionfrequency f₀. The dashed line represents results computed using equation(19) whereas the squares represent results obtained from micromagneticsimulation.

4. Advantages Over Conventional Inductive Coil Operation

The equations derived through theoretical analysis reveal a wealth ofinformation regarding the origin and nature of the advantages ofmagnetic field sensor operation based on RPM as well as providing ameans of quantifying some of these advantages. For example, equation (7)allows for the identification of two distinct sources from whichtransduced signal amplification may arise. The first of these sources,yielding the product term

$\left( \frac{M_{s}{\sin(\theta)}}{\Delta H} \right)$in equation (7), is the coupling of magnetization magnetic flux to theinductive detector, where amplification is achieved if this flux islarger than that which may be coupled directly due to the incidentmagnetic field. The second source, yielding the product term

$\frac{f_{0}}{f_{s}}$in equation (7), is the generation of magnetization terms of interest atf₀±f_(s) as a result of frequency mixing. With these frequenciestypically being much larger than the incident magnetic field frequencyf_(s), it follows immediately from the principles of Faraday inductionthat larger transduced signals will be produced, all else being equal.

As a whole, amplification can be more naturally understood from theinterpretation of RPM as a method of parametric amplification. In thiscontext, magnetization is the output of a resonant system dependent onthe parameter H_(e)(t), where the perturbing magnetic field H_(p)(t)behaves as the pump that harmonically drives the system and theelectromagnetic wave magnetic field h_(s)(t) serves to vary the systemparameter such that parametric amplification ensues. It is finally notedthat the sources of amplification have a remarkable elegance,originating directly from the magnetic material and intrinsic nonlinearnature of its magnetization dynamics, thus requiring no externalequipment or circuitry to achieve.

For an approximate quantification of attainable values, consider thecommonly used yttrium iron garnet (YIG) ferrite material biased suchthat f₀ is for example 1 GHz. At this frequency, it is reasonable toexpect the linewidth ΔH of the material to be around 0.2 Oe (Oersted).Then, for the optimistic upper limit case of a highly perturbed, or inother words strongly pumped, magnetization with θ of 90 degrees, signalamplification by a factor upwards of 4000 is reached due solely to thecoupling of magnetization magnetic flux. The total transduced signalamplification taking into account frequency mixing as well is then evenlarger and depends specifically on f_(s).

Intimately related to the amplification advantage of magnetic fieldsensors based on RPM is their advantage of a physically compact size,where size may be characterized by the size of the inductive detectioncoil. Simply put, equation (5) indicates that the area A of theinductive detection coil may be reduced by at most the value of equation(7) while still maintaining a transduced signal strength larger than orequal to that of a conventional inductive coil of the original size A.

Also closely related to the amplification advantage is the sensitivityadvantage of magnetic field sensors based on RPM. Again, the sources ofamplification originate directly from the magnetic material and itsintrinsic nonlinear magnetization dynamics. Distinct from conventionalcases where amplification is achieved by external circuitry after theantenna which raises not only the signal amplitude but also the noiselevel, the amplification offered by the RPM scheme occurs in thematerial before the inductive detection coil where Johnson thermal noiseand ferrite damping noise are generated, similar to the case of how alow noise amplifier is added before the antenna. Thus, transduced signalamplification is achieved without raising the noise generated by theinductive detection coil, immediately indicating a much-improvedsensitivity as compared to inductive antennas operating underconventional principles.

As mentioned previously, the limit to sensitivity is ultimatelydetermined by thermal noise generated by the ferrite loaded inductivedetection coil. For an approximate quantification of achievablesensitivities, a thin film YIG ferrite material with a linewidth of 0.2Oe, supposing a bias field of 70 Oe, corresponding to a thin filmresonance frequency of 1 GHz, then equation (10) indicates that a powerconsumption of just −24 dBm will allow for sub-pT/Hz^(1/2) sensitivity.

Of some merit is an explicit distinction between the advantages of RPMbased magnetic field sensors and those of the previously mentionedconventional antennas incorporating ferrites. Among these previousantennas, the ferrite rod antenna is of particular interest due to itsrelative success and widespread usage as well as its architecture, whichhas some initial resemblance to that of the RPM based magnetic fieldsensors. The ferrite rod antenna consists of a conducting coil wrappedaround a ferrite core, as illustrated in FIG. 2C, and operates entirelybased upon conventional direct voltage induction. The ferrite coreserves solely as a source of magnetization to magnify the magnetic fluxthat an incident magnetic field couples to the coil, likewise magnifyingthe transduced signal strength. In this regard, the ferrite rod antennapossesses the same advantages of amplification, size, and sensitivitycompared to a conventional inductive coil as does the RPM based magneticfield sensors.

However, not only is the typical amplification achieved by a ferrite rodinductive coil quite low, falling in the range of approximately a factorof 50, but also the antenna is associated with significant magneticlosses that do not affect the RPM based magnetic field sensors. Theselosses arise from the fact that ferrite core of the ferrite rodinductive coil is unbiased and includes hysteresis loss, loss due todomain wall resonance, and loss due to thermally activated domain wallmovements. The magnetic losses in the ferrite core become additionalsources of thermal noise for conventional ferrite rod antennas whilethey are absent in the RPM based sensors as the ferrite is biased intosaturation and free of domain wall movements. Furthermore, the loweramplification of the ferrite rod antenna immediately indicates lowersensitivity as well as lower potential size reduction as compared to theRPM based magnetic field sensors.

5. Considerations for Practical Implementation

For the purposes of tractability, a simplified scenario was maintainedthroughout much of the prior analyses and developments by making variousassumptions. While most of these assumptions are easily maintained inpractice, a particular few necessitate a thoughtful approach toimplementation and consideration of phenomena previously neglected inorder to hold to them. Perhaps chief amongst these phenomena isdemagnetization. Demagnetization is an equivalent description of thedipole-dipole interaction between subatomic magnetic moments of theferrite material and is in general dependent on the material shape,size, and magnetization orientation. Accounting for its effects onmagnetization dynamics may be accomplished with an additive termH_(d)(r,t) in the effective field of equation (1), written now as:H _(e)(r,t)=H ₀ +h _(s)(t)+H _(d)(r,t)  (11)where H_(d) (r,t) is known as the demagnetization field and both H₀ andh_(s)(t) are the familiar magnetic fields from prior analyses. Mostnotable about the demagnetization field in this context is the fact thatit is in general spatially nonuniform, thus clearly invalidating theoriginal assumption of a uniform H_(e) (r,t) and giving rise tocomplications in the practical implementation of RPM. The ramificationsof this nonuniformity can be approximately understood as a spatiallynonuniform magnetization precession resonance frequency which not onlydestroys the phase coherence of M(r,t) precession throughout theferrite, but also introduces additional complicating phenomena such asexchange coupling. The resulting information that may be extracted froma detection of precession during RPM operation is thus significantlydegraded in quality if not completely unusable.

The implementation approach taken to address the effects ofdemagnetization and maintain a uniform effective field involves anappropriate choice of ferrite shape. Specifically, the shape is selectedsuch that, for all orientations of a uniform M(t) occurring throughoutnormal RPM operation, H_(d) (r,t) is either negligible in magnitude orapproximately uniform in space. With several viable options that maysatisfy these conditions, in at least one example embodiment it has beenchosen to use ferrites with a thin film geometry for which in-planeorientations of M(t) will yield a negligible demagnetization fieldwhereas out of plane orientations will yield an approximately uniformdemagnetization field. Any arbitrary orientation of M(t) is effectivelya superposition of an in-plane and an out of plane component and thus itis clear that the uniformity of H_(e) (r,t) can always be maintained.Ferrites of a thin-film geometry, as compared to those of otherpotentially suitable shapes, have the additional benefit that associatedprocessing methods such as pulsed laser deposition, spin spray plating,or liquid phase epitaxy are mature enough to produce films of extremelyhigh quality.

With the complications accompanying demagnetization accounted for, allother details of practical implementation follow in a fairlystraightforward manner. In this work, the bias field H₀, required toestablish uniformity of M(t) as well as a precession center frequencyabout which the modulation of RPM occurs, is applied with permanentmagnets. The perturbation field H_(p)(t), required to maintain M(t)precession amid a finite magnetic loss, is applied using a conductingcoil driven continuously at f₀ of equation (2). Lastly, inductivedetection of magnetization dynamics, required to obtain an electricalsignal from which information may be extracted, is performed withanother conducting coil. With regards to this detection, it of interestto recognize that, since H_(d)(r,t) is not always negligible throughoutoperation of RPM using a thin film ferrite, equation (13) indicates thatit still has some effect on the magnetization precession dynamics. Thiseffect is a modification of the precession such that it approximatelyresembles a pendulum oscillation as visualized in FIG. 1D. While thishas no significant bearing on operation, it is of importance in decidingupon orientation of the inductive detector to achieve the largesttransduced signal.

FIG. 4 illustrates an example embodiment 150 of an RPM based magneticfield sensor implementation. In at least one embodiment a subset of thehardware was implemented with a thin film yttrium iron garnet on agadolinium gallium garnet substrate, with a loop used for inductivedetection of ferrite magnetization dynamics in which the loop is used toproduce the perturbation field and maintain magnetization precession.

A first signal generating source (e.g., function generator used in thetest) 152 is coupled to a perturbation loop 156 which is proximal to themain portion of the magnetic field sensor 160 with its thin film ferrite164 about which is an inductive detection loop 162, about which arepositioned biasing magnets 158. A transmitting coil 168 is seen coupledto a function generator 170 for simulating a signal source to bereceived. A circuit 166 receives the output from the inductive detectionloop and processes it to determine magnetic field information. In thefigure this circuit is exemplified as a signal analyzer for the sake oftesting.

FIG. 5 illustrates an example 190 of received power spectrum for anembodiment of the disclosure. The figure depicts the spectrum of powercoupled to the inductive detection loop. The solid line corresponds todriving the coil generating h_(s)(t) with a voltage of 45 mV and thedotted line corresponds to driving the coil with a voltage of 91 mV. Inboth cases, the coil is driven at a frequency of 30 kHz. It should beappreciated that these voltages and frequencies are provided by way ofexample and not limitation.

5.2. Micromagnetic Simulation

As a means of evaluating the validity of the RPM concept, the analyticalexpressions describing its operation, and the approach to its practicalimplementation, micromagnetic simulations are performed. Specifically,simulations were carried out using the Object Oriented MicromagneticFramework (OOMMF), which is a micromagnetic simulator widely used andwell recognized as the standard for accurate solutions.

Reproducing the scenario visualized in FIG. 2A, a ferrite material isbiased with a magnetic field H₀ of magnitude 50 Oe, resulting in aprecession resonance frequency f₀ of 848 MHz. An electromagnetic wavemagnetic field h_(s)(t) is applied as a sinusoid with various amplitudesat a frequency of 50 kHz. Lastly, a perturbing magnetic field H_(p)(t)is applied as a sinusoid with amplitude 0.025 Oe at a frequency of 848MHz. It should be noted that the amplitude of H_(p)(t) is somewhatarbitrary and need only to be sufficiently small such that it does notcontribute significantly to the effective field. The amplitudes andfrequency of h_(s)(t) are similarly arbitrary to an extent and need onlysatisfy the conditions for which equation (4) was derived. In accordancewith the approach to implementation taken to account for the effects ofdemagnetization, the ferrite is modeled to have a thin-film geometry, byway of example and not limitation, having dimensions 1×1×0.001 mm, withthe material itself having the properties of YIG and a Gilbert dampingcoefficient α of 1e-3.

The proposed approach is best applied to sensors with small form factorswith volume of less than 1 cm³, which can be easily fabricated with theexisting material fabrication approaches.

These test simulations were performed for a total time of 0.2milliseconds for each h_(s)(t) amplitude of interest from which thetransverse magnetization component M_(t)(t) is extracted and the ratioof M_(t) ^(f) ⁰ ^(±f) ^(s) to M_(t) ^(f) ⁰ is determined. This ratio isalso computed theoretically from equation (19) using the quality factorQ from equation (15) which is of unmodulated precession for a thin filmferrite, and a comparison with the simulation results seen plotted inFIG. 3 . The simulations not only demonstrate that the concept of RPM isfeasible and support the proposed approach to deal with the effects ofdemagnetization, but also FIG. 3 shows an excellent agreement betweensimulation and theory 3 which validates the developed analyticalexpressions.

5.3. Example Implementation

A magnetic field sensor operating based on RPM is implemented usingavailable hardware and materials. Results from experiments using thesensor provide further evidence supporting the feasibility of RPM basedoperation, validate the practical applicability of the developed RPMtheory, and are already able to demonstrate significant advantages overconventional inductive coil operation.

The experimental setup that was shown in FIG. 4 employs an epitaxiallygrown YIG thin film ferrite of dimensions 3.5×5×0.003 mm correspondingto a sensor volume of 0.0525 mm³. It will be appreciated that thematerial and dimensions are provided by way of example and differentferrite materials with different dimensions can be utilized withoutdeparting from the teachings of the present disclosure.

In the example described, inductive detection of the transversecomponent of magnetization M_(t)(t) was accomplished with a single turnloop of area 6 mm² constructed from a piece of copper soldered ontotraces fabricated on a Rogers 4003C board. Two neodymium permanentmagnets were used to apply the bias magnetic field H₀ with a magnitudeof approximately 70 Oe, corresponding to a measured precession resonancefrequency of approximately 1 GHz. The field h_(s)(t) is generated from a125-turn coil of area 3375 mm² positioned 9.9 cm from the center of theinductive detection loop and driven by a function generator. Lastly, theperturbation field H_(p)(t) is applied with a single turn loop of area95 mm² fabricated on a Rogers 4003C board and driven by a functiongenerator for the sake of this testing. For all results from this setup,the perturbation loop was driven sinusoidally at the precessionresonance frequency to deliver −55 dBm of power to the sensor whereasthe coil generating h_(s)(t) was driven sinusoidally with variedvoltages and frequencies to evaluate trends in the behavior of thesystem. It should be appreciated that the above provides examples usedin testing but do not otherwise limit the practice of the presentdisclosure.

In FIG. 5 is shown the power coupled to the inductive detection loop forthe cases of the coil generating h_(s)(t) driven with voltages of 91 mVand 45 mV across its terminals at a frequency of 30 kHz. In both cases,the expected sum and difference frequency terms f₀±f_(s) as well as theunmodulated frequency term f₀ are clearly evident, demonstrating thatRPM operation is achieved. Comparing between the two cases then it isseen that doubling the driving voltage effectively doubles the amplituderatio of equation (8). Recognizing that the driving voltage is directlyproportional to the h_(s)(t) field amplitude H_(s) by antenna theory,then the linear relationship between the amplitude ratios and H_(s) isthus confirmed.

Table 1, shown at the end of the specification, presents additionaldetailed data of this ratio, obtained by driving the coil with variouscombinations of voltages and frequencies, and provides a comparison withtheoretical expectations. The theoretical expectations were computedusing equation (8) where H₀, H_(s) and Q were determined throughprocedures detailed in a previous section. This table shows outstandingagreement between the computed values and those experimentally observed,solidifying the validity of the developed RPM theory as well as itsapplicability to a practical implementation.

For the purposes of comparison and assessment of the performance of theRPM based magnetic field sensor, the inductive detection loop of the RPMimplementation was also operated under conventional receiving antennaprinciples in a second experimental setup. This setup essentiallyinvolves the same incident magnetic field as in the first experimentalsetup now directly inducing voltages in the inductive detection loop inorder to achieve transduction into electrical signals. This field isgenerated by the same coil driven sinusoidally by a function generatorwith the same voltages and frequencies as previously exemplified. Thesame 9.9 cm distance separates the inductive detection loop from thecoil. Based on a comparison of the received signals observed to those ofthe first setup, an amplification metric is computed and presented inTable 2. From this table, it is seen that significant amplification isachieved for all cases under consideration, demonstrating promise forRPM based magnetic field sensor applications in the very low frequency(VLF) and low frequency (LF) bands. It is also possible to conclude fromTable 2 that amplification grows as the coil driving frequencydecreases, as expected from equation (7). It is further noted that theamplification can easily be increased with an increase in pumping power.

FIG. 6 illustrates an example embodiment 210 showing an experimentalsetup for measurement of sensitivity for the RPM based magnetic fieldsensor.

This setup resembles that of FIG. 4 in the configuration of the sensor218, the biasing magnets 220 (for static H₀), and the perturbation loop216 (generating h_(s)(t)), with the caveat that the distance between thecoil 226 (driven by transmitter generator source 228) and the inductivedetection loop 222 of sensor 218 in this example is 7.7 cm. However, thesetup of FIG. 4 as it stood was not suitable for measurements ofsensitivity, since the noise floor would be limited by phase noise ofthe function generator producing the perturbation field. Consequently,the received signals at the sum and difference frequencies f₀±f_(s) werecoherently demodulated down to baseband f_(s) using an Analog DevicesADL5380 evaluation board 232, shown with internal mixers 234 a, 234 band quadrature phase splitter 236.

The inductive detection loop 222 is connected to the radio frequencyport of the board 232 through an amplifier depicted as a 33 dB low noiseamplifier (LNA) 230. The perturbation loop 216 was driven by a functiongenerator 212 through a 2-way power splitter 214, with the other outputof the splitter connecting to the local oscillator port of theevaluation board 232. The in-phase output of the board 232 is connectedto a signal analyzer 242 through an LNA, such as 40 dB LNA 240, and thequadrature-phase output of the board is terminated with 50 Ohms 238.

For all results of this setup, the perturbation loop was drivensinusoidally at the precession resonance frequency of 1 GHz to deliver−40.96 dBm of power to the sensor whereas the coil generating h_(s)(t)was driven sinusoidally with a voltage of 0.45 mV across its terminalsat 10 kHz. This yielded a demodulated signal with a signal-to-noiseratio (SNR) of 41.5 dB, from which sensitivity may be computed to be27.1 pT/Hz^(1/2). It is noted that this sensitivity was found to befairly constant with varying frequency, as expected based on equation(20). Comparing conventional ferrite rod antennas operating at similarfrequencies, the ferrite rod antennas can achieve a sensitivity abouttwo to three orders of magnitude better than the disclosed RPM basedsensor, but they require a volume six orders of magnitude greater. Thetheoretical sensitivity was calculated using a modified version ofequation (20) that accounts for the hardware and electronics of thesetup, yielding a value of 15.7 pT/Hz^(1/2). The details of thiscalculation are presented in the methods section.

6. Summary Discussion

Magnetic field sensors based on RPM are introduced and their principaladvantages in amplification, sensitivity, and size as compared toreceiving antennas operating under conventional principles have beendemonstrated herein. The general concept of RPM operation has proven tobe viable through both simulation and experiment. With the RPM basedmagnetic field sensor implementation in this disclosure being of arather simple makeshift nature, it is expected that further refinementswill only establish the practical capabilities of these antennas moresubstantially. For example, an inductive detector design that mitigatesshielding effects on the ferrite material will result in a largertransduced signal as would alternatively or additionally provide anincrease in the ferrite material volume. Most promising about RPM basedmagnetic field sensor operation is the fact that it is fundamentallydifferent from conventional antenna operation, possessing the potentialto contribute a completely new paradigm to low frequency antenna designand overcome longstanding barriers in performance.

6.1 Methods

Theoretical amplitude ratio. The amplitude ratios of Table 1 werecomputed theoretically using equation (19). The value of H₀ was foundwith a DC gaussmeter. The values of H_(s) were found through measurementof the current through the coil producing the field of interest followedby application of the Biot-Savart law. The value of Q was computed asthe inverse of the 3 dB bandwidth of the loaded RPM sensor. Thisbandwidth was measured using a vector network analyzer under a 70 Oebiasing field. All values are provided in Table 4.

6.2 Theoretical Sensitivity

The theoretical sensitivity was computed using equation (21). The valueof ΔH was found using the definition ΔH=4παf₀/μ₀γ with the value of afound using equation (16), and the value of Q found as that of theunloaded RPM sensor. The sensor is matched to 50 Ohms, so this unloadedQ has a value twice that of the loaded Q. The value of P_(p) was foundbased on measurements using a vector network analyzer under a 70 Oebiasing field. All values are provided in Table 6.

7. Supplementary Notes

7.1 Supplementary Note 1

Inductive sensor sensitivity derivation and comparison.

The noise of an inductive sensor is dominated by Johnson thermal noiseproduced by the inductive detector. Following a procedure analogous tothe derivation of sensitivity for the RPM sensor, the sensitivity of theinductive sensor corresponds to the RMS field of interest H_(s)/√{squareroot over (2)}, where H_(s) is such thatV _(ind)=√{square root over (4kTR)}  (20)where R is the ohmic resistance of the inductive detector. Applyingequation (7) of the main manuscript and again recognizing thatsensitivity is typically expressed in terms of magnetic flux densitiesby convention, then equation (20) can be written asNAδB_(s,ind)ω_(s)=√{square root over (4kTR)}  (21)where NAδB_(s,ind) is the sensitivity. To also handle ferrite inductivesensors, an additional factor corresponding to the relative permeabilityof the ferrite μ_(r) is included in V_(ind), leading to the more generalexpressionμ_(r)NAδB_(s,ind)ω_(s)=√{square root over (4kTR)}  (22)

The sensitivity can be written explicitly from equation (22) as

$\begin{matrix}{{\delta\; B_{s,{ind}}} = \sqrt{\frac{kTR}{\mu_{r}^{2}N^{2}A^{2}\omega_{s}^{2}}}} & (23)\end{matrix}$and then made more practical for computation by employing the equations

$\begin{matrix}{L_{ind} = \frac{\mu_{r}\mu_{0}N^{2}A}{\iota}} & (24) \\{Q_{ind} = \frac{\omega_{s}L_{ind}}{R}} & (25)\end{matrix}$

These equations describe the coil inductance and quality factor at f_(s)respectively, where ι is the axial length of the coil. Substitutingequations (24) and (25) into equation (23) yields the final equation

$\begin{matrix}{{\delta\; B_{s,{ind}}} = \sqrt{\frac{4\mu_{0}kT}{\mu_{r}\omega_{s}vQ_{ind}}}} & (26)\end{matrix}$

An expression comparing the sensitivity of the inductive sensor to thatof the RPM sensor can then be obtained by taking the ratio of equation(26) to equation (21) to yield

$\begin{matrix}{{\frac{\delta\; B_{s,{ind}}}{\delta\; B_{s,{RPM}}} = {\frac{1}{\Delta H}\sqrt{\frac{2P_{p}}{\mu_{r}\mu_{0}{v\omega}_{s}Q_{ind}}}}}.} & (27)\end{matrix}$

7.2 Supplementary Note 2

7.2.1 Experimental Sensitivity Computation

The experimental sensitivity δB_(s,exp) is computed using the amplitudeH_(s) of the field of interest, a noise de-embedding factor X, and themeasured signal-to-noise ratio SNR using

$\begin{matrix}{{\delta\; B_{s,\exp}} = {X\frac{\mu_{0}H_{s}}{\sqrt{2SNR}}}} & (28)\end{matrix}$

The amplitude H_(s) is found by applying the Biot-Savart law as

$\begin{matrix}{H_{s} = {\frac{NI}{4\pi}{\int\frac{{\partial\iota^{\prime}} \times \overset{\hat{}}{r}}{r^{2}}}}} & (29)\end{matrix}$where N is the number of turns of the coil producing the field ofinterest, I is the current amplitude through the coil, r is the vectorfrom the coil to the field observation point, and at ∂ι′ is a length ofthe coil directed along the direction of current flow. The currentamplitude I is measured using a current probe and H_(s) is then computedusing equation (29) and the geometry of the coil.

The theoretical sensitivity of equation (21) corresponds to noisedominated by the sensor noise; however, noise of the second experimentalsetup is dominated by that of the first LNA connected to the RPM sensor.In order to compare experimental and theoretical sensitivities, the LNAnoise is de-embedded using the factor X, defined as

$\begin{matrix}{X = \frac{V_{n,{RPM}}}{\sqrt{V_{n,{RPM}}^{2} + V_{n,{LNA}}^{2}}}} & (30)\end{matrix}$where V_(n,RPM) is the RMS output noise voltage of the RPM sensor andV_(n,LNA) is the RMS input noise voltage of the first LNA. All valuesare provided in Table 5.

7.3 Supplementary Note 3

In FIG. 7 is shown a circuit model of the experimental RPM sensor, whereV₀ and V_(RPM) are the voltages as defined in equation (9), and wherethe component values are provided in Table 3.

Embodiments of the present technology may be described herein withreference to flowchart illustrations of methods and systems according toembodiments of the technology, and/or procedures, algorithms, steps,operations, formulae, or other computational depictions, which may alsobe implemented as computer program products. In this regard, each blockor step of a flowchart, and combinations of blocks (and/or steps) in aflowchart, as well as any procedure, algorithm, step, operation,formula, or computational depiction can be implemented by various means,such as hardware, firmware, and/or software including one or morecomputer program instructions embodied in computer-readable programcode. As will be appreciated, any such computer program instructions maybe executed by one or more computer processors, including withoutlimitation a general purpose computer or special purpose computer, orother programmable processing apparatus to produce a machine, such thatthe computer program instructions which execute on the computerprocessor) or other programmable processing apparatus create means forimplementing the function) specified.

Accordingly, blocks of the flowcharts, and procedures, algorithms,steps, operations, formulae, or computational depictions describedherein support combinations of means for performing the specifiedfunction), combinations of steps for performing the specified function),and computer program instructions, such as embodied in computer-readableprogram code logic means, for performing the specified function). Itwill also be understood that each block of the flowchart illustrations,as well as any procedures, algorithms, steps, operations, formulae, orcomputational depictions and combinations thereof described herein, canbe implemented by special purpose hardware-based computer systems whichperform the specified function) or step), or combinations of specialpurpose hardware and computer-readable program code.

Furthermore, these computer program instructions, such as embodied incomputer-readable program code, may also be stored in one or morecomputer-readable memory or memory devices that can direct a computerprocessor or other programmable processing apparatus to function in aparticular manner, such that the instructions stored in thecomputer-readable memory or memory devices produce an article ofmanufacture including instruction means which implement the functionspecified in the block) of the flowchart). The computer programinstructions may also be executed by a computer processor or otherprogrammable processing apparatus to cause a series of operational stepsto be performed on the computer processor or other programmableprocessing apparatus to produce a computer-implemented process such thatthe instructions which execute on the computer processor or otherprogrammable processing apparatus provide steps for implementing thefunctions specified in the block) of the flowchart), procedure)algorithm), step), operation), formula(e), or computational depiction).

It will further be appreciated that the terms “programming” or “programexecutable” as used herein refer to one or more instructions that can beexecuted by one or more computer processors to perform one or morefunctions as described herein. The instructions can be embodied insoftware, in firmware, or in a combination of software and firmware. Theinstructions can be stored local to the device in non-transitory media,or can be stored remotely such as on a server, or all or a portion ofthe instructions can be stored locally and remotely. Instructions storedremotely can be downloaded (pushed) to the device by user initiation, orautomatically based on one or more factors.

It will further be appreciated that as used herein, that the termsprocessor, hardware processor, computer processor, central processingunit (CPU), and computer are used synonymously to denote a devicecapable of executing the instructions and communicating withinput/output interfaces and/or peripheral devices, and that the termsprocessor, hardware processor, computer processor, CPU, and computer areintended to encompass single or multiple devices, single core andmulticore devices, and variations thereof.

From the description herein, it will be appreciated that the presentdisclosure encompasses multiple implementations of the technology whichinclude, but are not limited to, the following:

A method comprising detecting a weak very low frequency (VLF) magneticfield based on spin precession modulation (SPM) of electron resonancefrequency or resonant precession modulation (RPM).

A method for detecting a weak very low frequency (VLF) magnetic fieldbased on spin precession modulation (SPM) of electron resonancefrequency or resonant precession modulation (RPM), the methodcomprising: (a) designating a biasing direction; (b) adding a dynamicVLF magnetic field along the biasing direction wherein the electronresonance frequency is modulated, and wherein a frequency modulatedsignal is generated with a spectrum that has two primary sidetonesbeside an original electron resonance spectral line; (c) wherein thesidetones contain both amplitude and phase information of the originalweak VLF field a carrier frequency upconverted by the resonance; and (d)detecting one of the two sidetones with a pickup coil wherein a highersensitivity results compared to detecting the original VLF field with acoil.

A very low frequency (VLF) antenna apparatus based on spin precessionmodulation (SPM) of electron resonance frequency or resonant precessionmodulation (RPM).

An apparatus for detecting weak very low frequency (VLF) signals basedon spin precession modulation (SPM) of electron resonance frequency orresonant precession modulation (RPM), the antenna comprising: (a) first,second and third loop antennas; (b) the loop first and second loopantennas oriented to be co-axial and spaced apart; (c) a magnetic spinsource positioned within the first loop antenna; (d) a magnetic biasingsource; (e) a pump source coupled to the second loop antenna; and (f)the third loop antenna configured for coupling to a VLF field.

An apparatus for detecting weak very low frequency (VLF) signals basedon spin precession modulation (SPM) of electron resonance frequency orresonant precession modulation (RPM), the antenna comprising: (a) firstand second loop antennas; (b) the loop antennas oriented to be co-axialand spaced apart; (c) a magnetic biasing source; (d) a pump sourcecoupled to the second loop antenna; and (e) the first loop antennaconfigured for coupling to a VLF field.

An apparatus for detecting weak very low frequency (VLF) signals basedon spin precession modulation (SPM) of electron resonance frequency orresonant precession modulation (RPM), the antenna comprising: (a) first,second and third loop antennas; (b) the loop first and second loopantennas oriented to be co-axial and spaced apart; (c) a magnetic spinsource positioned within the first loop antenna; (d) a magnetic biasingsource; (e) a pump source coupled to the second loop antenna; and (f)the third loop antenna configured for coupling to a VLF field.

The apparatus or method of any preceding implementation, furthercomprising: (a) designating the biasing direction as the Z-direction;(b) wherein the biasing field and initial magnetization is along theZ-direction; (c) applying a circularly polarized microwave pulse with alength of π/2 at the frequency of the ferromagnetic resonance frequencywith a pair of orthogonally placed coils and rotating the magnetizationby 90 degrees into the X-Y plane; (d) removing the microwave pulsewherein free precession begins; and (e) during free precession, addingthe VLF field along the biasing direction wherein the VLF fieldmodulates the magnetization precession and is captured with saidorthogonally placed set of coils.

The apparatus or method of any preceding implementation, wherein saidapparatus is configured for performing steps comprising detecting a weakvery low frequency (VLF) magnetic field based on spin precessionmodulation (SPM) of electron resonance frequency or resonant precessionmodulation (RPM).

The apparatus or method of any preceding implementation, wherein saidapparatus is configured for performing steps comprising: (a) designatinga biasing direction; (b) adding a dynamic VLF magnetic field along thebiasing direction wherein the electron resonance frequency is modulated,and wherein a frequency modulated signal is generated with a spectrumthat has two primary sidetones beside an original electron resonancespectral line; (c) wherein the sidetones contain both amplitude andphase information of the original weak VLF field a carrier frequencyupconverted by the resonance; and (d) detecting one of the two sidetoneswith a pickup coil wherein a higher sensitivity results compared todetecting the original VLF field with a coil.

The apparatus or method of any preceding implementation, wherein saidapparatus is configured for performing steps comprising: (a) designatingthe biasing direction as the Z-direction; (b) wherein the biasing fieldand initial magnetization is along the Z-direction; (c) applying acircularly polarized microwave pulse with a length of π/2 at thefrequency of the ferromagnetic resonance frequency with a pair oforthogonally placed coils and rotating the magnetization by 90 degreesinto the X-Y plane; (d) removing the microwave pulse wherein freeprecession begins; and (e) during free precession, adding the VLF fieldalong the biasing direction wherein the VLF field modulates themagnetization precession and is captured with said orthogonally placedset of coils.

The apparatus or method of any preceding implementation, wherein saidapparatus is configured for performing steps comprising detecting a weakvery low frequency (VLF) magnetic field based on spin precessionmodulation (SPM) of electron resonance frequency or resonant precessionmodulation (RPM).

Implementations include each and every embodiment of the technologydescribed herein, as well as any aspect, component, or element of anyembodiment described herein, and any combination of aspects, componentsor elements of any embodiment described herein.

As used herein, term “implementation” is intended to include, withoutlimitation, embodiments, examples, or other forms of practicing thetechnology described herein.

As used herein, the singular terms “a,” “an,” and “the” may includeplural referents unless the context clearly dictates otherwise.Reference to an object in the singular is not intended to mean “one andonly one” unless explicitly so stated, but rather “one or more.”

Phrasing constructs, such as “A, B and/or C”, within the presentdisclosure describe where either A, B, or C can be present, or anycombination of items A, B and C. Phrasing constructs indicating, such as“at least one of” followed by listing a group of elements, indicatesthat at least one of these group elements is present, which includes anypossible combination of the listed elements as applicable.

References in this disclosure referring to “an embodiment”, “at leastone embodiment” or similar embodiment wording indicates that aparticular feature, structure, or characteristic described in connectionwith a described embodiment is included in at least one embodiment ofthe present disclosure. Thus, these various embodiment phrases are notnecessarily all referring to the same embodiment, or to a specificembodiment which differs from all the other embodiments being described.The embodiment phrasing should be construed to mean that the particularfeatures, structures, or characteristics of a given embodiment may becombined in any suitable manner in one or more embodiments of thedisclosed apparatus, system or method.

As used herein, the term “set” refers to a collection of one or moreobjects. Thus, for example, a set of objects can include a single objector multiple objects.

Relational terms such as first and second, top and bottom, and the likemay be used solely to distinguish one entity or action from anotherentity or action without necessarily requiring or implying any actualsuch relationship or order between such entities or actions.

The terms “comprises,” “comprising,” “has,” “having,” “includes,”“including,” “contains,” “containing” or any other variation thereof,are intended to cover a non-exclusive inclusion, such that a process,method, article, or apparatus that comprises, has, includes, contains alist of elements does not include only those elements but may includeother elements not expressly listed or inherent to such process, method,article, or apparatus. An element proceeded by “comprises . . . a”, “has. . . a”, “includes . . . a”, “contains . . . a” does not, without moreconstraints, preclude the existence of additional identical elements inthe process, method, article, or apparatus that comprises, has,includes, contains the element.

As used herein, the terms “approximately”, “approximate”,“substantially”, “essentially”, and “about”, or any other versionthereof, are used to describe and account for small variations. Whenused in conjunction with an event or circumstance, the terms can referto instances in which the event or circumstance occurs precisely as wellas instances in which the event or circumstance occurs to a closeapproximation. When used in conjunction with a numerical value, theterms can refer to a range of variation of less than or equal to ±10% ofthat numerical value, such as less than or equal to ±5%, less than orequal to ±4%, less than or equal to ±3%, less than or equal to ±2%, lessthan or equal to ±1%, less than or equal to ±0.5%, less than or equal to±0.1%, or less than or equal to ±0.05%. For example, “substantially”aligned can refer to a range of angular variation of less than or equalto +10°, such as less than or equal to 5°, less than or equal to ±4°,less than or equal to ±3°, less than or equal to 2°, less than or equalto 1°, less than or equal to 0.5°, less than or equal to ±0.1°, or lessthan or equal to ±0.05°.

Additionally, amounts, ratios, and other numerical values may sometimesbe presented herein in a range format. It is to be understood that suchrange format is used for convenience and brevity and should beunderstood flexibly to include numerical values explicitly specified aslimits of a range, but also to include all individual numerical valuesor sub-ranges encompassed within that range as if each numerical valueand sub-range is explicitly specified. For example, a ratio in the rangeof about 1 to about 200 should be understood to include the explicitlyrecited limits of about 1 and about 200, but also to include individualratios such as about 2, about 3, and about 4, and sub-ranges such asabout 10 to about 50, about 20 to about 100, and so forth.

The term “coupled” as used herein is defined as connected, although notnecessarily directly and not necessarily mechanically. A device orstructure that is “configured” in a certain way is configured in atleast that way, but may also be configured in ways that are not listed.

Benefits, advantages, solutions to problems, and any element) that maycause any benefit, advantage, or solution to occur or become morepronounced are not to be construed as a critical, required, or essentialfeatures or elements of the technology describes herein or any or allthe claims.

In addition, in the foregoing disclosure various features may groupedtogether in various embodiments for the purpose of streamlining thedisclosure. This method of disclosure is not to be interpreted asreflecting an intention that the claimed embodiments require morefeatures than are expressly recited in each claim. Inventive subjectmatter can lie in less than all features of a single disclosedembodiment.

The abstract of the disclosure is provided to allow the reader toquickly ascertain the nature of the technical disclosure. It issubmitted with the understanding that it will not be used to interpretor limit the scope or meaning of the claims.

It will be appreciated that the practice of some jurisdictions mayrequire deletion of one or more portions of the disclosure after thatapplication is filed. Accordingly the reader should consult theapplication as filed for the original content of the disclosure. Anydeletion of content of the disclosure should not be construed as adisclaimer, forfeiture or dedication to the public of any subject matterof the application as originally filed.

The following claims are hereby incorporated into the disclosure, witheach claim standing on its own as a separately claimed subject matter.

Although the description herein contains many details, these should notbe construed as limiting the scope of the disclosure but as merelyproviding illustrations of some of the presently preferred embodiments.Therefore, it will be appreciated that the scope of the disclosure fullyencompasses other embodiments which may become obvious to those skilledin the art.

All structural and functional equivalents to the elements of thedisclosed embodiments that are known to those of ordinary skill in theart are expressly incorporated herein by reference and are intended tobe encompassed by the present claims. Furthermore, no element,component, or method step in the present disclosure is intended to bededicated to the public regardless of whether the element, component, ormethod step is explicitly recited in the claims. No claim element hereinis to be construed as a “means plus function” element unless the elementis expressly recited using the phrase “means for”. No claim elementherein is to be construed as a “step plus function” element unless theelement is expressly recited using the phrase “step for”.

TABLE 1 Amplitude Ratio Results and Theory Experimental TheoreticalVoltage Frequency Ratio Ratio (mV) (kHz) (dB) (dB) 45 10 −47.31 −46.8930 −56.82 −56.64 50 −61.91 −61.08 91 10 −41.44 −40.87 30 −50.96 −50.6250 −55.52 −55.06

TABLE 2 Amplification Results Frequency (kHz) Amplification (dB) 10+22.19 30 +17.00 50 +13.37

TABLE 3 Component Values for Circuit Model (FIG. 7) of RPM Sensor Biasedfor a 710 MHz Resonance Frequency Component Description Value L_(f)Inductance of detector feeding 6.1883 nH L Inductance of detector 0.7640nH R₀ Ohmic resistance of detector 1.0717 Ω C Capacitance of detector0.1287 pF R_(r) Radiation resistance 3.4124 Ω L_(m) Effective inductanceof ferrite 0.1485 nH R_(m) Effective resistance of ferrite 118.96 ΩC_(m) Effective capacitance of ferrite 0.3380 nF

TABLE 4 Theoretical Amplitude Ratio Calculation Variable Values VariableDescription Value H₀ Biasing field 70 Oe M_(s) Saturation 140 kA/mmagnetization Q Unloaded quality factor 135.95 H_(s) Signal field atvarious voltages and frequencies Voltage Freq. 1 V 10 kHz 357.1 mA/m 30kHz 116.2 mA/m 50 kHz 69.7 mA/m 2 V 10 kHz 714.2 mA/m 30 kHz 232.5 mA/m50 kHz 139.4 mA/m

TABLE 5 Sensitivity Calculation Variable Values Variable DescriptionValue I Current Amplitude 61.82 uA H_(s) Field of Interest amplitude8.59 nT SNR Signal-to-noise ratio 41.51 dB X De-embedding factor 0.531V_(n, RPM) RPM sensor output noise voltage 0.895 nV/Hz^(1/2) V_(n, LNA)LNA input noise voltage 1.429 nV/Hz^(1/2)

TABLE 6 Theoretical Sensitivity Calculation Variable Values VariableDescription Value H₀ Biasing field 70 Oe M_(s) Saturation magnetization140 kA/m ΔH Linewidth 0.496 Oe Q Unloaded quality factor 271.9 P_(p)Pumping power −40.957 dBm T Temperature 290 K

What is claimed is:
 1. A method for detecting a weak very low frequency(VLF) magnetic field, the method comprising: (a) placing biasing magnetsproximal to a ferrite material to establish a bias magnetic field in abiasing direction; (b) generating a perturbing magnetic field along thebiasing direction of the ferrite material, wherein said perturbingmagnetic field is generated at a frequency to maintain magnetizationprecession in the ferrite material with a constant angle from itsequilibrium state amid the effects of damping; (c) detecting themagnetic field of the ferrite material by an inductive detection loop;and (d) wherein, in response to receiving the weak very low frequency(VLF) magnetic field as an incident wave, said ferrite materialgenerates terms in the precession dynamics at sum and differencefrequencies between resonant precession frequency of the ferritematerial and frequency of the incident wave being sensed as an amplifiedfrequency modulated signal having signal amplification which arises fromintrinsic non-linear magnetization dynamics of the ferrite materialprior to being sensed by the inductive detection loop, withmagnetization of the ferrite material then picked up by the inductivedetection loop, and generating an electrical output whose spectrum hassum and difference frequencies as well as an electron resonancefrequency of the ferrite material, wherein the sum and differencefrequencies contain both amplitude and phase information of the incidentwave to be sensed by the inductive detection loop; and (e) detecting oneof the sum and difference frequencies with the inductive detection loopoperating at electron spin resonance frequency, as a frequencymodulation of the precession from which both magnitude and phaseinformation about the weak very low frequency (VLF) magnetic field isdetermined, and whereby an increase in sensitivity is provided whencompared to detecting the weak VLF field with only the inductivedetection loop without the combination of the ferrite material, thebiasing field, and the perturbing magnetic field.
 2. The method of claim1, further comprising: (a) wherein the biasing direction is in aZ-direction; (b) wherein the biasing field and initial magnetization isalong the Z-direction; (c) applying a circularly polarized microwavepulse with a length of Π/2 at a frequency of ferromagnetic resonancefrequency with a pair of orthogonally placed coils and rotatingmagnetization by 90 degrees into an X-Y plane; (d) removing themicrowave pulse wherein free magnetization precession begins; and (e)during free precession, adding the perturbing magnetic field along thebiasing direction wherein the perturbing magnetic field modulates themagnetization precession and is captured with said orthogonally placedset of coils.
 3. A very low frequency (VLF) antenna apparatuscomprising: (a) a ferrite material proximal to which are placed biasingmagnets to establish a bias magnetic field in a biasing direction; (b) asignal generating source coupled to an inductive detection loop forgenerating a perturbing magnetic field along the biasing direction ofthe ferrite material to maintain magnetization precession in the ferritematerial with a constant angle from its equilibrium state amid theeffects of damping; (c) an antenna configured as the inductive detectorloop for magnetic field detection of the ferrite material; (d) detectingone of sum and difference frequencies with the antenna operating at anelectron spin resonance frequency of the ferrite material, as afrequency modulation of the precession from which both magnitude andphase information about the weak very low frequency (VLF) magnetic fieldis determined, and whereby an increase in sensitivity is provided whencompared to detecting the weak VLF field with only the inductivedetection loop without being in combination with the ferrite material,the biasing field, and the perturbing magnetic field.
 4. A very lowfrequency (VLF) antenna apparatus comprising: (a) a ferrite materialproximal to which are placed biasing magnets to establish a biasingdirection in a Z-direction, which create a biasing field and initialmagnetization along the Z-direction; (b) orthogonally placed coils inrelation to said ferrite material; (c) a pulse generator configured forapplying a circularly polarized microwave pulse with a length of Π/2 ata frequency of ferromagnetic resonance of the ferrite material, with theorthogonally placed coils and which performs rotating the magnetizationby 90 degrees into an X-Y plane; (d) removing the microwave pulsewherein free precession begins; and (e) during free precession,generating the VLF field along the biasing direction wherein the VLFfield modulates magnetization precession as said ferrite materialgenerates terms in precession dynamics at sum and difference frequenciesbetween resonant precession frequency of the ferrite material andfrequency of incident wave being sensed as an amplified frequencymodulated signal having signal amplification which arises from intrinsicnon-linear magnetization dynamics of the ferrite material and iscaptured with said orthogonally placed coils which detect the sum and/ordifference frequencies which result from frequency modulation of theprecession from which both magnitude and phase information about theweak very low frequency (VLF) magnetic field is determined.